David A. Reid with Christine Knipping

Proof in Mathem

atics Education

Proof in Mathematics Education

Research, Learning and Teaching

David A. Reid

with

Christine KnippingAcadia University, Wolfville, Canada

Research on teaching and learning proof and proving has expanded in recent decades. This refl ects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education. This development is a welcome one for those interested in the topic, but also poses a challenge, especially to teachers and new scholars. It has become more and more diffi cult to get an overview of the fi eld and to identify the key concepts used in research on proof and proving.

This book is intended to help teachers, researchers and graduate students to overcome the diffi culty of getting an overview of research on proof and proving. It reviews the key fi ndings and concepts in research on proof and proving, and embeds them in a contextual frame that allows the reader to make sense of the sometimes contradictory statements found in the literature. It also provides examples from current research that explore how larger patterns in reasoning and argumentation provide insight into teaching and learning.

S e n s e P u b l i s h e r s DIVS

Proof in Mathematics EducationResearch, Learning and Teaching

David A. Reid with Christine Knipping

S e n s e P u b l i s h e r s

Proof in Mathematics Education

Proof in Mathematics Education

Research, Learning and Teaching David A. Reid Christine Knipping Acadia University, Wolfville, Canada

SENSE PUBLISHERS ROTTERDAM/BOSTON/TAIPEI

A C.I.P. record for this book is available from the Library of Congress. ISBN: 978-94-6091-244-3 (paperback) ISBN: 978-94-6091-245-0 (hardback) ISBN: 978-94-6091-246-7 (e-book) Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands http://www.sensepublishers.com Printed on acid-free paper All Rights Reserved © 2010 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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TABLE OF CONTENTS

Introduction........................................................................................................... xiii Part I: What is Proof? 1. History of Proof ................................................................................................... 3

The Standard View ............................................................................................ 3 Other Views of the History of Proof................................................................ 10 Summary ......................................................................................................... 24

2. Usages of “Proof ” and “Proving” ..................................................................... 25

Everyday Usages ............................................................................................. 25 Scientific Usages ............................................................................................. 26 Mathematical Usages ...................................................................................... 26 Usages in Mathematics Education Research ................................................... 27 “Demonstration” and “Proof ” in Other Languages......................................... 32 Summary ......................................................................................................... 33

3. Researcher Perspectives..................................................................................... 35

Philosophies of Mathematics........................................................................... 37 Research Based on an a Priorist Philosophy of Mathematics ......................... 39 Research Based on an Infallibilist Philosophy of Mathematics....................... 40 Research Based on a Quasi-Empiricist Philosophy of Mathematics ............... 46 Research Based on a Social-Constructivist Philosophy of Mathematics......... 48 Summary ......................................................................................................... 52 Balacheff’s Epistemologies of Proof ............................................................... 53 Diverse or Comprehensive Perspectives?........................................................ 54

Part 2: Important Research Foci, Past and Present 4. Empirical Results.............................................................................................. 59

Key Studies...................................................................................................... 59 Many Students Accept Examples as Verification ............................................ 59 Many Students Do Not Accept Deductive Proofs as Verification ................... 62 Many Students Do Not Accept Counterexamples as Refutation ..................... 63 Students Accept Flawed Deductive Proofs as Verification.............................. 64 Students’ Criteria for the Acceptance of Arguments ....................................... 65 Students Offer Empirical Arguments to Verify................................................ 67 Most Students Cannot Write Correct Proofs.................................................... 68 Ideas for Research ........................................................................................... 70

5. The Role of Proof ............................................................................................. 73

The Roles of Proof in Mathematics ................................................................. 73

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What is Proving in School Mathematics? ...................................................... 79 The Roles of Proof for Students..................................................................... 80 Possible Roles of Proof in Teaching .............................................................. 81 Ideas for Research.......................................................................................... 82

6. Types of Reasoning ......................................................................................... 83

Deductive Reasoning ..................................................................................... 84 Inductive Reasoning ...................................................................................... 88 Abductive Reasoning..................................................................................... 99 Reasoning by Analogy ..................................................................................110 Other Kinds of Reasoning ........................................................................... 123 Summary...................................................................................................... 126 Ideas for research......................................................................................... 127

7. Classifying Proofs and Arguments ................................................................ 129

Proofs and Arguments Described According to the Representations Involved ....................................................................................................... 130 Other Classifications of Proofs and Arguments ........................................... 142 Ideas for Research........................................................................................ 151

8. Argumentation............................................................................................... 153

Argumentation Versus Proof........................................................................ 155 Argumentation in Accord with Proof........................................................... 158 Argumentation According to Krummheuer ................................................. 161 Summary...................................................................................................... 163 Argumentation in Japan ............................................................................... 164 Ideas for Research........................................................................................ 164

9. Teaching Experiments ................................................................................... 165

Fawcett......................................................................................................... 165 The Debate Approach .................................................................................. 169 Expecting Explanations ............................................................................... 172 Italy.............................................................................................................. 173 Summary...................................................................................................... 175 Ideas for Research........................................................................................ 176

Part 3: Processes of Reasoning and Argumentation 10. Argumentation Structures.............................................................................. 179

Toulmin’s Functional Model and Argumentation Structures ....................... 179 The Source-Structure ................................................................................... 180 The Reservoir-Structure............................................................................... 185 The Spiral-Structure..................................................................................... 187 The Gathering-Structure .............................................................................. 189 Ideas for Research ....................................................................................... 191

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11. Patterns of Reasoning .................................................................................... 193 Deduce-Conjecture-Test cycle ..................................................................... 194 Proof Analysis.............................................................................................. 198 Scientific Verification .................................................................................. 201 Surrender...................................................................................................... 202 Exception and Monster Barring ................................................................... 204 Summary...................................................................................................... 207 Ideas for Research........................................................................................ 208

Part 4: Conclusions 12. Implications for Teaching...............................................................................211

What is Proof and what is it for? ..................................................................211 Formality...................................................................................................... 212 Results from New Math and Two-Column Proof Teaching ......................... 215 Starting where Students and Teachers are.................................................... 217 Teaching Experiments.................................................................................. 218 Summary...................................................................................................... 219

13. Directions for Research ................................................................................. 221

Teaching Proof............................................................................................. 221 Students’ Understandings of Proof............................................................... 223 Conceptional Issues ..................................................................................... 224 Conclusion ................................................................................................... 225

References............................................................................................................ 227 Author Index ........................................................................................................ 241 Subject Index........................................................................................................ 245

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LIST OF FIGURES

Figure 1: A misleading diagram showing that 8×8 = 5×13 ................................. 14 Figure 2: Word usage in three ESM papers ......................................................... 29 Figure 3: Three dimensions of description for researcher perspectives............... 36 Figure 4: Diagram from Fischbein, 1982, p. 18................................................... 40 Figure 5: Diagram that formed the basis for the triangle angle sum proof

in Fawcett, 1938................................................................................... 42 Figure 6: Reid’s PRISM model............................................................................ 58 Figure 7: The Count the Squares problem ........................................................... 86 Figure 8: Drawing used by Bill and John when proving that the sum of two

odd numbers is even............................................................................. 86 Figure 9: Will’s Count the Squares pattern.......................................................... 94 Figure 10: Diagram used in French classroom for Pythagorean Theorem

proof................................................................................................... 104 Figure 11: Handshake diagram for six people .................................................... 105 Figure 12: Sofia’s diagram .................................................................................. 106 Figure 13: The relationship between analogy, generalisation and

specialisation.......................................................................................113 Figure 14: The Arithmagon problem (as posed by Reid, 1995b)..........................115 Figure 15: Triangles showing geometric properties drawn by Wayne

while exploring ...................................................................................116 Figure 16: A tetrahedron with equal masses at its vertices from Hanna and

Jahnke, 2002b..................................................................................... 122 Figure 17: Toulmin model ................................................................................... 180 Figure 18: Overall argumentation structure in the proving process in

Mr. Lüders’ class ............................................................................... 182 Figure 19: Proof diagram from Lüders’ class ...................................................... 182 Figure 20: Argumentation stream AS-4 from Lüders’ class ................................ 183 Figure 21: Proof diagram from Nissen’s class..................................................... 184 Figure 22: The source-structure in Nissen’s class ............................................... 185 Figure 23: The reservoir-structure in Pascal’s class............................................. 186 Figure 24: Diagram used in Pascal’s class for Pythagorean Theorem

proof................................................................................................... 186 Figure 25: Overall argumentation structure in Dupont’s class............................. 187 Figure 26: Argumentation structure in James’ class for the rhombus

proving process .................................................................................. 188 Figure 27: Argumentation structure in James’ class for the side-side-side

proving process .................................................................................. 190 Figure 28: The Deduce-Conjecture-Test cycle .................................................... 194 Figure 29: The first Arithmagon puzzle............................................................... 194 Figure 30: Sandy’s first guess: 5.......................................................................... 195 Figure 31: Sandy’s solution to the first puzzle..................................................... 195

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Figure 32: Sandy’s symbols for the unknowns and givens in a general Arithmagon ........................................................................................ 196

Figure 33: Sandy’s puzzle.................................................................................... 199 Figure 34: Sandy’s reasoning, including Proof Analysis ..................................... 201 Figure 35: Scientific Verification......................................................................... 202 Figure 36: Surrender............................................................................................ 202 Figure 37: Exception and Monster Barring ......................................................... 205

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LIST OF PROOFS

Proof 1: Elements Book I Proposition 1............................................................... 5 Proof 2: Figure reconstructed on the basis of Liu Hui’s commentary on

Jiuzhang Suanshu ................................................................................ 12 Proof 3: Elements Book I Proposition 4............................................................. 17 Proof 4: Elements Book IX Proposition 20........................................................ 19 Proof 5: Proof that the sum of the interior angles of a triangle is

180 degrees .......................................................................................... 44 Proof 6: Textbook proof from Chazan, 1993, p. 365.......................................... 96 Proof 7: Empirical argument from Chazan, 1993, p. 366................................... 97 Proof 8: The sum of the first n integers (by MI) ................................................ 99 Proof 9: The perpendicular bisectors of a triangle meet in a point....................119 Proof 10: The medians meet in a single point, physics proof ........................... 122 Proof 11: The Goldbach conjecture ................................................................... 131 Proof 12: Product of negatives .......................................................................... 132 Proof 13: Try it with 15...................................................................................... 132 Proof 14: The diagonals of a rhombus are congruent ........................................ 133 Proof 15: There are 14 possibilities and all fit ................................................... 134 Proof 16: Not all prime numbers are odd........................................................... 134 Proof 17: Numeric Gauss proof ......................................................................... 134 Proof 18: Divisibility by Nine............................................................................ 135 Proof 19: Action proof of the commutativity of multiplication ......................... 136 Proof 20: Behold!............................................................................................... 137 Proof 21: Schorle proof...................................................................................... 138 Proof 22: Two-column proof for the triangle angle sum.................................... 139 Proof 23: Symbolic Gauss proof ........................................................................ 140 Proof 24: Infinitude of primes............................................................................ 140 Proof 25: The product of two diagonal matrices is diagonal.............................. 140 Proof 26: Proof of an algebraic identity ............................................................. 141 Proof 27: A formal proof.................................................................................... 142 Proof 28: A transformational proof .................................................................... 148

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LIST OF TABLES

Table 1: References in Euclid’s proof of Prop. I.1 ............................................... 6 Table 2: Chronology of people and publications mentioned in the text ............... 8 Table 3: Summary of research perspectives ....................................................... 53 Table 4: Balacheff’s epistemologies of proof..................................................... 54 Table 5: Key studies reviewed ........................................................................... 60 Table 6: Results showing that many students accept examples as

verification. .......................................................................................... 61 Table 7: Summary of results showing that many students and teachers

do not accept deductive proofs as verification ..................................... 63 Table 8: Summary of results showing students accept flawed deductive

proofs as verification............................................................................ 65 Table 9: Factors influencing acceptance of arguments....................................... 66 Table 10: Frequencies of ratings for the familiar and unfamiliar

deductive verifications ......................................................................... 66 Table 11: Students’ use of empirical arguments................................................... 68 Table 12: Students able to write correct proofs .................................................... 69 Table 13: Students who were able to construct a valid proof

for TIMSS item K18 ............................................................................ 69 Table 14: Results according to the type and truth value of the statements........... 70 Table 15: Reasoning dichotomies based on certainty .......................................... 90 Table 16: Types of abductive reasoning used in examples................................. 103 Table 17: Bill’s analogy ......................................................................................112 Table 18: Ben’s analogy......................................................................................117 Table 19: Some links between triangles and tetrahedra ..................................... 120 Table 20: Kinds of reasoning ............................................................................. 126 Table 21: Kinds of reasoning and roles.............................................................. 127 Table 22: Comparison of Balacheff and the preformalists’ classifications of

arguments ........................................................................................... 129 Table 23: Classification of arguments ................................................................ 131 Table 24: Overview of categories ...................................................................... 143 Table 25: Comparison of proof classification systems....................................... 144 Table 26: Harel and Sowder’s proof schemes ................................................... 149 Table 27: Notions of argumentation................................................................... 163

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INTRODUCTION

Research on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education. This development is a welcome one for those interested in the topic, but also poses a challenge, especially to teachers and new scholars. It has become more and more difficult to get an overview of the field and to identify the key concepts used in research on proof and proving.

When we met, Christine was working on her doctoral dissertation (Knipping, 2003b). She commented on the difficulty she had making sense of the existing research. David understood this feeling having had the same struggle when working on his doctoral dissertation (Reid, 1995b). In the interval the amount of research to be read and understood had increased, but the relationship between the work of different researchers was no more apparent. Key terms were used differ-ently by different authors, disparate theoretical assumptions were made, phenomena were classified in incompatible ways, all without comment. We wished that some synthesis of the literature existed that would explain the discrepancies and make the links we found missing. And having achieved a better understanding ourselves of the literature as a result of our efforts, we wondered if we could attempt such a synthesis, perhaps in a journal article. In our discussions it soon became clear that a longer piece of writing would be needed, and this book is the outcome.

This book is intended to help teachers, researchers and students to overcome the difficulty of getting an overview of research on proof and proving. It reviews the key findings and concepts in research on proof and proving, and embeds them in a contextual frame that allows the reader to make sense of the sometimes contradictory statements found in the literature.

The first part provides this frame. It begins with an outline of the history of proof in mathematics, both as it is usually presented and as it is interpreted by scholars who take a wider view. Then the various uses of the words “proof ” and “proving” in everyday life, science, mathematics and mathematics education are described and compared. Finally, the various perspectives taken by researchers in the field are outlined and placed into a structure that allows for comparison.

The second part reviews current research. First, basic findings from empirical research are summarised. Then important theoretical constructs and classification systems are discussed in several chapters organised around the themes of the role of proof, reasoning, types of proof, and argumentation. Finally, several teaching experiments are described.

The third part focusses on two larger frameworks for examining proving and argumentation. The first is argumentation processes which are social processes that occur in classrooms (and elsewhere) through which knowledge changes status. A method of describing and analysing argumentation processes is outlined which

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reveals differences in the role of argumentation in different contexts. The second is reasoning processes, which are internal psychological processes which link together different ways of reasoning about mathematics. Distinctive patterns in reasoning processes are described and related to the goals of teaching proof.

The final part includes concluding comments, first on the implications research on proof has for teaching, and then on questions that require further research. Throughout there is an emphasis on exploring the multiple perspectives different researchers bring to the study of proof in mathematics education. These perspect-ives are seen as being not only inevitable in a field where international attention is brought to problems that often have significant local elements, but also enriching as a diversity of perspectives offers opportunities to make sense of phenomena that might be seen in a limited way from a single perspective. Hence, we do not attempt a combining of perspectives, as Harel and Fuller (2009) have done. While it can be confusing to encounter multiple perspectives, we believe it is also very valuable. We hope this book will decrease the confusion and increase the rewards of its readers in their further explorations of proof in mathematics education.

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PART I

WHAT IS PROOF?

Reading any research literature can be a challenge at first because most authors make assumptions about what the reader already knows about the field. This is necessary as there is never space to include all the background underlying a publication. In the research literature on teaching and learning proof, assumptions are often made about the historical context of proof in mathematics, the meanings of words like “proof ” and “proving” and about the theoretical perspective of the author, which is often assumed to be shared by the reader. In Part 1 we consider these three sets of assumptions and provide a guide to what assumptions might be made by authors in the field. Unfortunately, but perhaps necessarily, there is not a single uniform set of assumptions all researchers on proof share. Hence, we will describe a range of possibilities, without being able to state definitively what assumptions a given publication is based on. Given an outline of the possibilities, however, a reader should be able to pick up on the clues in a publication and identify the assumptions being made.

Chapter 1 concerns the history of mathematics, and presents an outline of the “standard” history of proof in mathematics, familiarity with which is often assumed when proof is discussed. We also present several alternatives to key elements in the standard history, that some authors refer to in their work.

Chapter 2 discusses the uses of the words “proof ” and “proving” in mathematics, mathematics education, logic, science and everyday life. Authors sometimes write from more than one of these perspectives, which means that their terminology can shift meaning from one paragraph to the next. Being aware of the possible contexts and the meanings for “proof ” and “proving” associated with them will help readers find their way through this shifting terrain.

Chapter 3 explores the theoretical perspectives of researchers on proof in mathe-matics education. From within a given perspective, it seems a natural way of seeing things, and so authors often do not comment on the perspective they take. However, for communication with the larger community some awareness of these perspectives is necessary, and for a reader new to the field understanding that different pers-pectives exists will aid in making sense of what sometimes seem to be contradictory statements.

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CHAPTER 1

HISTORY OF PROOF

Before embarking on a discussion of proof and proving in mathematics education, a look back at proof and proving in the history of mathematics is in order. This will provide the necessary background to some of the issues we will discuss in later chapters, and introduce some important concepts related to the nature of proof, and the acceptance of proofs.

A student of modern mathematics might be confused at this point, as her or his experience of mathematical proof in school might have suggested that mathematics grows by an accumulation of knowledge, so although there are now new theorems that have been proven since the time of the Greeks, the word “proof ” in the context of mathematics has meant the same thing since the time of Euclid, at least. However, as Wilder (1981) points out: “‘proof ’ in mathematics is a culturally determined, relative matter. What constitutes proof for one generation, fails to meet the standards of the next or some later generation” (p. 40). By exploring this variation we can discover other ways of perceiving proof, and other ways of proving.

As you read this chapter you may want to reflect on this question: – Does the history of proof in mathematics have direct implications for the

teaching and learning of proof? If so, how?

THE STANDARD VIEW

When one reads a history of mathematics (e.g., Anglin, 1994; R. Jones, 1997; Kleiner, 1991; Kline, 1962), one is likely to encounter a version of the history of proof we call the “standard” view. When the history of mathematics is mentioned in research on teaching and learning proof, it is usually the standard view that is assumed, and so it has had significant impacts on proof teaching and research. In this section we will summarise the standard view. In the next section we will introduce some critiques and alternatives.

The First Proofs

According to the standard view, proofs originated with the Greeks, specifically with Thales (c. 600 BCE). Prior to that time mathematics was done without proofs. A number of theorems are associated with Thales, not because he discovered them, but because he proved them:

All these theorems were known to the Egyptians and Mesopotamians. The reason they are associated with Thales is that he was the first person to offer proofs for them. This was an essential difference between pre-Greek and Greek

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mathematics: the Greeks established the logical connections among their results, deducing the theorems from a small set of starting assumptions or axioms. (Anglin, 1994, p. 14)

A number of authors have speculated on the reason the Greeks began to insist on proving mathematical statements. Some (e.g., Hannaford, 1998, p. 181; Kleiner, 1991, p. 293) have claimed that the democratic nature of Athenian society created a context in which logical argument was valued. Others have noted that the existence of a leisure class meant that there were individuals who had time for philosophical and mathematical activity without any immediate practical application (e.g., Kline, 1962, p. 45). Kleiner (1991, p. 293) and Arsac (2007, p. 31) also mention the problem of the incommensurability of the side and diagonal of a square, and Kleiner adds the need to teach mathematics, as motivations for an emphasis on proof. Hanna and Barbeau (2002) see the motivation for proving as arising from the nature of the entities studied in mathematics:

For the early Egyptians, Babylonians, and Chinese, the weight of observational evidence was enough to justify mathematical statements. But classical Greek mathematicians found this way of determining mathematical truth or false-hood less than satisfactory. They saw that, unlike other sciences, mathematics often deals with entities that are infinite in extent or number, such as the set of all natural numbers, or are abstractions, such as triangles or circles. When dealing with such entities, mathematics needs to make absolute statements, that is, statements that apply to every instance without exception. (p. 36)

Whatever the reason, the origin of mathematical proofs is credited to the Greeks, whose innovation then spread to other cultures.

The Legacy of Logic

The standard view tells us that rather than basing mathematics on observation and experiment, the Greeks based it on logic. Not only did they make use of logical arguments, they reflected on their reasoning and their methods. Plato (427–349 BCE) argued for a restriction on the tools that could be used in geometry, and Aristotle (384–322 BCE) formulated the methods appropriate to mathematics:

In the Posterior Analytics, Aristotle formulates what we call the deductive method. It was adopted by Euclid and has always been an essential charac-teristic of mathematics. This method consists of starting with propositions called axioms and then proving propositions called theorems. Each statement in the proof has to be justified either by an axiom or by a previously proved theorem or by a principle of logic. (Anglin, 1994, p. 63)

For example, consider Proof 1, which is the first proof in Heath’s (1956) translation of Euclid’s Elements (c. 300 BCE). The text consists of several parts: The proposition to be proven (line 2), a description of what has to be proven given a specific case (the segment AB in lines 3–4), a construction (lines 5–12), a proof that the object constructed is what it is meant to be (lines 13–20), and finally a statement asserting that what has been done is what was required (lines 21–23).

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Both the steps of the construction and the proof are justified by references to common notions, postulates and definitions that are stated earlier in the Elements (see Table 1 for those referred to in Proof 1). Euclid’s “common notions” and “postulates” are assumptions that are to be accepted without justification. Nowadays they would usually both be called “axioms”. Euclid’s distinction between them is that common notions apply outside of geometry, while his postulates are specific to geometry. Euclid’s Elements is a structured presentation of the mathematics of that time. He did not discover any of the theorems he presented, but he did present them as part of a larger structure. The Elements provided the model for proof in mathematics, and in other domains, for centuries.

Euclid’s contribution was the logical organisation of the Elements – its axiomatic structure in which everything is carefully deduced from a small number of definitions and assumptions. This structure served as a model for Aquinas’s Summa Contra Gentiles, for Newton’s Principia, and for Spinoza’s Ethics. The Elements has been the most influential textbook in history. (Anglin, 1994, p. 81)

Proof 1: Elements Book I Proposition 1

Proposition 1. On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construct an equilateral

triangle on the straight line AB.

5 With centre A and distance AB let the circle BCD be described; [Post. 3] again, with centre B and distance BA

let the circle ACE be described; [Post. 3] and from the point C, in which the

10 circles cut one another, to the points A, B let the straight lines CA, CB be

joined. [Post. 1] Now, since the point A is the centre of

the circle CDB, AC is equal to AB. 15 [Def. 15] Again, since the point B is the centre of the circle CAE, BC is equal to BA.

[Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are also equal to one another; [C.N. 1]

therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal 20 to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite

straight line AB. (Being) what it was required to do.

Heath, 1956, Vol. 1, pp. 241–242, line numbers adjusted

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Table 1. References in Euclid’s proof of Prop. I.1

Common Notion 1 Things which are equal to the same thing are also equal to one another.

Postulate 1 Let the following be postulated: To draw a straight line from any point to any point.

Postulate 3 To describe a circle with any centre and distance.

Definition 15 A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.

Euclid’s axiomatic approach is not only of historical importance. It has become a central motif in mathematics. Modern mathematical structures are often systems based on definition and axioms and relying on rules of inference. ... In such a system, a proposition is considered true if it can be derived from the axioms in a finite number of logical steps using the permitted rules of inference. (Hanna & Barbeau, 2002, p. 38)

Descartes (1596–1650) found the inspiration for his philosophical method in the Elements. What impressed him was the way Euclid based geometry on a few axioms and proceeded to deduce further statements about which one could have absolute confidence.

Those long chains of reasoning, each of them simple and easy, that geometri-cians commonly use to attain their most difficult demonstrations, have given me an occasion for imagining that all the things that can fall within human knowledge follow one another in the same way and that, provided only that one abstain from accepting anything as true that is not true, and that one always maintains the order to be followed in deducing the one from the other, there is nothing so far distant that one cannot finally reach nor so hidden that one cannot discover. (Descartes, 1637/1993, p. 11)

For thinkers from Aristotle and Descartes to the present day, the deductive method is associated with certainty.

Euclid regarded his starting assumptions not as mere hypotheses, but as truths. He intended to instantiate the ideal described by Aristotle as the beginning of the Posterior Analytics: sure basic knowledge is obtained by the rigorous deduction of the consequences of basic truths. These truths are either definitions or existence assertions. (Anglin, 1994, p. 82)

Of all those who have already searched for truth in the sciences, only the mathematicians were able to find demonstrations, that is, certain and evident reasons. (Descartes, 1637/1993, p. 11)

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The Restoration of Rigour

In the standard view European mathematics since the Renaissance is a continuation of the work of the Greeks. True, there were times when the discovery of new theorems and methods overtook the task of rigourously proving them, for example, when Newton and Leibniz introduced the calculus in the seventeenth century. Their justifications for their methods were criticised as not up to the standard of Euclid by, for example, Berkeley. This difficulty was addressed finally in the nineteenth century, by Cauchy and others (see Kleiner, 1991, for an outline of this history and references to other sources).

At the same time criticisms of lack of rigour in analysis were being addressed, an important development occurred in the history of geometry: the invention of non-Euclidean geometries. For Descartes and Kant, Euclidean geometry was an example of knowledge that was undeniably true. Its foundation was the nature of space itself. When Lobachevsky, Bolyai and Gauss announced that it was possible to construct a geometry in which one of Euclid’s postulates (the famous parallel postulate) is false, it became possible to question the truth of Euclidean geometry. Of course, there was strong temptation to assume there was something wrong with non-Euclidean geometry, that there was a contradiction somewhere. In 1871 Klein eliminated this possibility by proving that if there is a contradiction in the new non-Euclidean geometries then there is also a contradiction in Euclidean geometry. This created the need for a new approach to securing the foundations of geometry and the rest of mathematics, as it was not longer possible to convincingly claim that Euclidean geometry was the true geometry of space (Many histories of mathematics tell this story. Kline, 1972, is thorough.).

This method of establishing the lack of contradiction (the ‘consistency’) of one mathematical system by showing it is just as consistent as another system, was applied not only to geometry. Hilbert showed that the various geometries were as consistent as basic arithmetic, raising the question of the consistency of arithmetic. The next step was to try to establish arithmetic on the basis of set theory, and then to establish set theory on the basis of logic (the logicist approach of Russell, Whitehead, etc.). In the late nineteenth and early twentieth centuries this led to a new focus on axiomatisation and the axiomatisation of set theory (by Frege, Russell and others), geometry (by Hilbert) and arithmetic (by Peano and others).

The next step was to replace traditional mathematical statements with purely formal statements that could themselves be the objects of calculation. This was the objective of Hilbert’s formalism. Mathematics, from a formalist perspective, is the manipulation of symbols, without any reference to any meaning or interpretation.

Mathematical proof will consist of this process: the assertion of some formula; the assertion that this formula implies another; the assertion of the second formula. A sequence of such steps in which the asserted formulas or the implications are proceeding axioms or conclusions will constitute the proof of a theorem. Also, substitution of one symbol for another or a group of

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symbols is a permissible operation. Thus formulas are derived by applying the rules for manipulating the symbols of previously established formulas. (Kline, 1972, p. 1205)

Some key authors and publications in the history of mathematics, and the history of thought more generally are listed in Table 2, to provide an overview of the standard version of the history of proof as we have presented it here. Note that there is a considerable gap between the work of Aristotle and Euclid and the European philosophers and mathematicians who found those works significant. Authors and works who are often not discussed when relating the standard history of proof are omitted here, resulting in this gap. We are not claiming that no mathematics happened in this period, only that it is usually ignored when the standard version is presented. Some Chinese works we will discuss later are included as a hint that the standard version may be incomplete.

Table 2. Chronology of people and publications mentioned in the text

600 BCE Thales, first historically recorded proofs 427–349 BCE Plato 384–322 BCE Aristotle (335 BCE Posterior Analytics) 300 BCE Euclid’s Elements 250 BCE–100 CE

Jiuzhang Suanshu (exact date of composition uncertain, and parts may be much older)

0 200 263 Liu Hui’s commentary on the Jiuzhang Suanshu 300–600 Theoretical phase in Chinese mathematics 800 1000 1200 1258–1264 Thomas Aquinas’s Summa Contra Gentiles composed 1400 1500–1557 Tartaglia 1596–1650 Descartes (1637 Method) 1632–77 Spinoza (1677 Ethics published posthumously) 1642–1727 Newton (1687 Philosophiæ Naturalis Principia Mathematica) 1646–1716 Leibniz 1685–1753 Berkeley (1734 The Analyst) 1724–1804 Kant 1789–1857 Cauchy 1830 Bolyai and Lobachevsky publish first treatises on non-Euclidean geometry 1845–1918 Cantor 1848–1925 Frege 1849–1925 Klein (1871 Proof that a contradiction in non-Euclidean geometry

implies a contradiction in Euclidean geometry) 1858–1932 Peano 1862–1943 Hilbert (1899 Foundations of Geometry) 1872–1970 Russell (1910–1913 Principia Mathematica, with Whitehead)

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Degrees of formality. It is unlikely that the reader will have encountered many proofs that meet the formalist definition of mathematical proof (an example is Proof 27 in Chapter 7). However many proofs are called “formal” that do not meet the formalist definition. This can be confusing when trying to interpret a statement like “students of mathematics should understand formal proof ” (Moore, 1990, p. 57).

Lakatos (1978) distinguishes between three different degrees of formality in proofs: pre-formal, formal, and post-formal. Lakatos uses “formal” in the same sense as the formalists: A sequence of symbols that makes it possible to “mechanically decide of any given alleged proof if it really was a proof or not” (p. 62). By “pre-formal” he means a proof which is accepted as such by mathematicians, convincing, but not a formal proof.

[In a pre-formal proof] there are no postulates, no well-defined underlying logic, there does not seem to be any feasible way to formalize this reasoning. What we were doing was intuitively showing that the theorem was true. This is a very common way of establishing mathematical facts, as mathematicians now say. The Greeks called this process deikmyne and I shall call it thought experiment. (pp. 64–65)

A similar description has been adopted by a group of mathematics education researchers we refer to as “preformalists”. We will discuss their work in Chapters 3 and 7. Note, however, that Lakatos refers specifically to informal proofs acceptable to mathematicians, and the preformalists include proofs that could be made acceptable. To avoid confusion we will follow the preformalists’ usage. We will use “semi-formal” to refer to informal proofs acceptable to mathematicians, instead of Lakatos’s term “pre-formal” and use the word “preformal” (without a hyphen) to refer to the proofs discussed by the preformalists.

Lakatos’s “post-formal” proofs are proofs about formal proofs. For example, the proof of the Duality Principle in Projective Geometry “Although projective geometry is a fully axiomatized system, we cannot specify the axioms and rules used to prove the Principle of Duality, as the meta-theory involved is informal.” (p. 68). Other examples include the consistency and completeness proofs of formal systems such as Gödel’s proof of his Incompleteness Theorem.

Most proofs are either preformal or semi-formal. However, the influence of the formalists has made proofs in general more formal, and within the sub-disciplines of mathematical foundations and computer science formal proofs are the norm.

In the early years of the twentieth century the mathematical community began to have confidence that the formal structures they were developing would, for mathe-matics at least, achieve what Leibniz had dreamed of in the eighteenth century, “an exhaustive collection of logical forms of reasoning—a calculus ratiocinator—which would permit any possible deductions from initial principles” (Kline, 1980, p. 183). The standard history of proof suggests that this has, in fact, been achieved.

It is believed that every mathematical text can be formalised. Indeed, it is believed that every mathematical text can be formalised within a single formal language. This language is the language of formal set theory. (Davis & Hersh, 1981, p. 136)

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By formalising mathematics, it was possible to revise the proofs of the past to new standards of rigour inspired by, but improving upon, Euclid’s Elements.

During the period from about 1821 to 1908 ... mathematicians restored and surpassed the standards of rigour which had been established during the period of classical Greek mathematics. (R. Jones, 1996)

Formalist work at the foundations of mathematics inspired the Bourbaki group in France to apply axiomatic approaches to algebra and analysis, which in turn inspired some of the reforms of the New Math curriculum reforms of the 1960s. This brings us to the present day. In the standard view today’s proofs are direct descendants of the proofs of Euclid, although today’s proofs make more use of symbols to make formalisation easier. Like Euclid’s proofs they start from axioms and lead to results that are “true” within the structure defined by the axioms.

OTHER VIEWS OF THE HISTORY OF PROOF

Not everyone accepts the standard view of the history of proof, and alternative viewpoints have emerged that challenge many claims of the standard view. These challenges are usually based on historical evidence and sociological analyses. In this section we will discuss the challenges to several claims in standard view of the history of proof, including these: – Proof began in Greece and was limited to cultures with an intellectual connection

to Greece, primarily those in Europe. “For the early Egyptians, Babylonians, and Chinese, the weight of observational evidence was enough to justify mathe-matical statements” (Hanna & Barbeau, 2002, p. 36). If one accepts that “the deductive method. ... has always been an essential characteristic of mathematics” (Anglin, 1994, p. 63) then one must conclude that what the early Egyptians, Babylonians, and Chinese did was not mathematics.

– In Euclid’s Elements “everything is carefully deduced from a small number of definitions and assumptions” (Anglin, 1994, p. 81). Euclid’s proofs are models of mathematical rigour.

– The work of Russell, Frege, etc. re-established mathematics on firm foundations. In principle, every mathematical proof can be reduced to a sequence of formal statements, in which each statement follows from previous statements according to the rules of symbolic logic.

– Proofs transmit truth from established axioms to the theorems they prove. The purpose of proofs is to make this connection from the axioms to the theorems. We will consider each of these beliefs in turn, but first, it is important to note a

feature of the history of mathematics that makes any discussion of specific practices problematic.

The history of mathematics is spread over a wide time period and a wide range of cultures, and in many cases the data available is far from ideal. In the cases of Greek and Chinese mathematics, the original sources are lost, and most of what we know about them comes from sources written a thousand years after the originals. In the case of the Greek texts the copies we have come from Arab sources that had their own rich mathematical culture, which may have influenced the transcription and

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translation of the texts they had available. We can see this process in more recent cases; for example, in Heath’s translation of Euclid’s Elements from Greek to English his footnotes indicate places where he chose to translate passages into formulations more accessible to contemporary readers, but differing from the original Greek text (this occurs often; see, for example, the footnotes to Book I Proposition 4, Heath, 1956, p. 248). Aside from purposeful changes to the texts, there are also accidental changes and missing sections that have forced later translators, transcribers, and historians to interpolate material that might not be the same as in the original. With Egyptian and Mesopotamian sources, we are a bit better off in that some original papyri and cuneiform tablets have survived, but the interpretation of them is a challenge for experts as the original languages fell into disuse and had to be reconstructed.

In addition, we have no way of knowing if the mathematical texts we have from these cultures are representative of their mathematical practices. If one plucked a book about mathematics at random from all those printed in the twenty-first century, the likelihood is that one would end up with a school book, as many more school books are printed than university texts or specialists’ monographs. Such a school book would hardly represent the present level of development of mathematics.

The lack of original data is one problem facing historians of mathematics, but the diversity of the data is another. Anyone attempting to look at the whole picture is forced to work from secondary or even tertiary sources, as the number of academics with strong backgrounds in both mathematics and history, and able to read Arabic, traditional Chinese, ancient Egyptian, classical Greek, Sanskrit and Sumerian (to name only a few of the languages in which important mathematical texts have been written) is probably very small. As Smoryński (2008) comments:

The further removed from the primary, the less reliable the source: errors are made and propagated in copying; editing and summarising can omit relevant details, and replace facts by interpretations; and speculations can become established fact even though there is no evidence supporting the “fact”. (p. 11)

These considerations alone should make one cautious of accepting the standard view of the history of proof (or any view of the history of proof), and there are also some other reasons to be wary.

Proof in China

The standard view of the history of proof claims that proof originated in Greece, and that while Chinese mathematics includes many significant discoveries, the Chinese did not prove. “Mikami [1913] considered the greatest deficiency in old Chinese mathematical thought was the absence of the idea of rigorous proof ” (Needham, 1959, p. 151). Since the 1960s, however, Western scholars have been aware that there is at least one work of ancient Chinese mathematics in which proofs play an important role:

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[While] most Chinese mathematical works contain no justifications ... there is one major exception, namely a set of Chinese argumentative discourses which has been handed down to us from the first millennium AD. We are essentially referring to the commentaries and sub-commentaries of the Jiuzhang Suanshu, the key work which inaugurated Chinese mathematics and served as a reference for it over a long period of its history. (Martzloff, 1997, p. 69)

The Jiuzhang Suanshu (Computational Prescriptions in Nine Chapters) is the oldest Chinese mathematical text known to us, having been compiled beginning in 200 BCE (Martzloff, 1997, p. 124). The proofs in the commentaries by Liu Hui were made at the end of the third century, at the beginning of a significant period in Chinese mathematics:

From the third to the sixth century, Chinese mathematics entered its theoretical phase. For the first time, it seems, importance was attached to proofs in their own right, to the extent that trouble was taken to record these in writing. Approximate values for the number π were then derived by computation and reasoning, rather than simply via an empirical process. (p. 14)

Proof 2: Figure reconstructed on the basis of Liu Hui’s commentary on Jiuzhang Suanshu

A right triangle has sides of 8 steps and 15 steps. What is the diameter of its inscribed circle? Compute the Xian from the Gou and the Gu, then add the three together and divide this sum into twice the product of the Gou and the Gu. [Find c from a and b (using the Pythagorean theorem), then add a + b + c and divide by 2ab (the area of the two figures to the right)]

adapted from Siu, 1993, p. 350 and Martzloff, 1989, p. 150, whose reconstructions are based on that of Li Huang (?–1812)

Area = Twice the product of the Gou and the Gu = 2ab

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It must be said, however, that Liu Hui’s proofs are not like Euclid’s proofs. For one thing, Euclid expressed his proofs primarily through words, but Chinese mathe-maticians made extensive use of diagrams. For example, consider Proof 2, which shows a visual proof based on the text of Liu Hui’s commentary on problem 16 of Chapter 9 of the Jiuzhang Suanshu. The original diagrams are, unfortunately, lost. Siu (1993) suggests a reconstruction of the figures, which we have changed slightly to make the argument clearer. In modern terms, the theorem proven is:

Given a right triangle with sides a, b, c, the diameter α of the inscribed circle is 2ab / (a + b + c).

Note that the solution is given in general terms (the words “Gou” and “Gu” are used to refer to the shorter and longer legs of the triangle, instead of using the numbers given in the problem) and that it is assumed that the reader knows how to calculate the length of the hypotenuse (the Xian) from the lengths of the legs (i.e., that the reader is familiar with the “Pythagorean” theorem).

This practice of basing proofs on visually convincing diagrams continued when Euclid’s Elements was translated into Chinese after it was introduced by Jesuit missionaries. For example, Mei Wending (1633–1721) made changes to Euclid’s diagrams when he incorporated parts of the Elements into his Jihe bubian (Comple-ments of Geometry).

He modified the figures to make them immediately readable, although Euclid operated in the opposite direction, thus making it necessary to resort to deductive reasoning. (Martzloff, 1997, p. 113)

This preference for readable figures over verbal descriptions is one reason why Chinese proofs are still not accepted as proofs by some historians of mathematics (Siu, 1993, p. 345).

The use of diagrams is sometimes rejected entirely and misleading diagrams are given to support a claim that basing proofs on diagrams is not reliable.

The prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important — but they prove nothing. (Brown, 1999, p. 25)

Philosophers and mathematicians have long worried about diagrams in mathe-matical reasoning — and rightly so. They can indeed be highly misleading. (p. 43)

One such misleading diagram is shown in Figure 1. There are also other reasons beyond the use of diagrams for the perception of

the rarity of proofs in Chinese mathematics. Perhaps the most important is the role of Chinese mathematical texts as textbooks in established schools.

Under the Sui dynasty (518–617), and above all under the Tang dynasty (618–907), mathematics was officially taught at the guozixue (School for the Sons of the State), based on a set of contemporary or ancient textbooks as written support. (Martzloff, 1997, p. 15)

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Figure 1. A misleading diagram showing that 8×8 = 5×13.

Indeed, to a first approximation, a majority of Chinese mathematical works may be better represented as pedagogical tools, in other words, didactic aids used to teach numerical computation, together with prescriptive texts (for example, user manuals). (p. 47)

As with problem based textbooks today, the intent was that the students should work out the proofs for themselves. In his commentary on Jiuzhang Suanshu, Liu Hui “associates in the same sentence two famous passages from the Lunyu (Confucian Analects) which both suggest an idea of the same order:”

I told him what had gone before and he understood what followed: [I] showed him a corner [i.e. an aspect of the question] and he replied with the other three.

Here the author indicates his wish not to disclose all the details of his reasoning to the student.... Consequently, instead of giving the details of his own thought processes, he often merely indicates that the solution of a given problem is analogous to that of some other problem, or that “the remainder follows in the same way” (Martzloff, 1997, p. 70, bracketed insertions in quotation from original)

Anyone who has studied from a contemporary university mathematics textbook will be familiar with hints like “the remainder follows in the same way.” They often appear in exercises in which new theorems are stated without their proofs, which are left to be discovered by the students. Herbst (2002b) describes how such exercises came to be included in high school geometry textbooks in the late nineteenth century.

A characteristic of Chinese culture that may also have affected the nature of Chinese proofs was the emphasis in Chinese literary style on conciseness.

Such a knowledge excluded literary ornaments and excessively long passages such as those found in Euclidean theorems and proofs. In particular, syllogisms and other logical forms were especially unacceptable for they involved numerous

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repetitions and redundancies: they were contradictory with the canons of Chinese literary redaction which, in the case of technical subjects, valued conciseness above all. (Martzloff, 1997, p. 112)

It is also worth remembering that in Western countries there have been times when mathematicians, far from publishing proofs of their theorems, kept their results secret (for example, Tartaglia). The same phenomenon occurred in China:

Last but not least, assuming that an inventor had succeeded in creating novel procedures, it is not certain that he would have been inclined ipso facto to reveal the secrets of their creation; in fact, the existence of rivalry between calendarist astronomers is known. (Martzloff, 1997, p. 49)

In summary, the belief that the practice of proving mathematical results began in Greece and spread from there to other, primarily European, cultures, is a myth. Proving was also a part of Chinese mathematics from the third century, and possibly earlier, but in a different style than Euclid’s. That many historians based a belief that proof was not a part of Chinese mathematics on their lack of awareness of Liu Hui’s commentaries is a useful reminder that absence of evidence is not the same thing as evidence of absence. This might suggest that we approach with caution the belief that proof was not a part of mathematics in other cultures where significant mathematical discoveries were made (for example, Egypt, Mesopotamia, and India). As an illustration of this point we will now briefly consider proof in India before moving on the to the claim that Euclid’s proofs are models of rigour.

Proof in India

The standard history of mathematics makes claims about proof in India that are similar to those made about China. For example, in describing Hindu mathematics in the period 200–1200 CE Kline (1972) writes:

There is much good procedure and technical facility, but no evidence that they considered proof at all. They had rules, but apparently no logical scruples. (p. 190)

Joseph (1992, 1994) critiques this claim, pointing out that, as in China, proofs (“upapatti”) were often included in commentaries on mathematical texts, even if they were not a part of the texts themselves. There is, however, an important difference between upapatti and Euclid’s proofs:

The upapattis of Indian mathematics are presented in a precise language, displaying the steps of the argument and indicating the general principles which are employed. In this sense they are no different from the “proofs” found in modern mathematics. But what is peculiar to the upapattis is that while presen-ting the argument in an “informal” manner (which is common in many mathe-matical discourses anyway), they make no reference whatsoever to any fixed set of axioms or link the given argument to “formal deductions” performable with the aid of such axioms. (1992, p. 194; see also 1994, p. 200)

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It is this lack of reference to axioms that distinguishes the upapattis of India from Euclid’s proofs. This is related, as Joseph notes, to a focus on rigour and certainty that is important to the Euclidean approach as opposed to an emphasis on understanding and clarity that is present in the Indian tradition. We will encounter similar differences in the role of proof in subsequent chapters (especially Chapter 5). Now, however, we will turn to the question of whether Euclid’s proofs are as rigourous and his theorems are established with certainty as the standard history of proof claims.

The Euclid Myth

In the standard view Euclid’s proofs are taken to be models of mathematical rigour, that establish theorems with certainty. While there is no denying that Euclid’s Elements has had a profound influence on Western mathematics, the claim of certainty has been questioned:

What is the Euclid myth? It is the belief that the books of Euclid contain truths about the universe which are clear and indubitable. Starting from self-evident truths, and proceeding by rigorous proof, Euclid arrives at knowledge which is certain, objective, and eternal. Even now, it seems that most educated people believe in the Euclid myth. Up to the middle of the nineteenth century, the myth was unchallenged. Everyone believed it. (Davis & Hersh, 1981, p. 325)

There are two aspects to this “myth”. One is the assertion that Euclid’s proofs are rigourous, and the second that the knowledge arrived at using the deductive method is certain and objective. Here we will discuss the first of these aspects, rigour. Later we will consider the second aspect in the context of twentieth century mathematics.

Rigour in Euclid’s proofs. Do the proofs in the Elements live up to the claims sometimes made about them, that “everything is carefully deduced from a small number of definitions and assumptions” (Anglin, 1994, p. 81)? In fact, they do not. Euclid’s proofs make use of assumptions that are never stated, some involve reference to physical manipulations (as in Liu Hui’s proofs) and some use specific cases to justify general conclusions.

Since Euclid still has popularity, and even with mathematicians, a reputation for rigour in virtue of which his circumlocution and longwindedness are condoned, it may be worth while to point out, to begin with, a few of the errors in his first twenty-six propositions. (Russell, 1903/1937, p. 404)

Recall, for example, the first proof in Book I, the construction of an equilateral triangle (see Proof 1, on page 5). Each step in the construction (lines 5–12) indicates the postulate that states that such a construction is possible, and each step in the proof (lines 13–22) indicates which definition, postulate or common notion justifies that step in the argument.

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Notice, in line 9, the mention of “the point C, in which the circles cut one another.” From the diagram it is clear that there are two such points. That Euclid seems to claim that there is only one is perhaps a minor flaw. He is only trying to prove that it is possible to construct an equilateral triangle; that the construction might produce two does not make it invalid.

More significantly, Euclid does not provide a common notion, postulate, or definition to let us know when we can actually construct a “point C, in which the circles cut one another.” While these circles intersect, it has not been established

Proof 3: Elements Book I Proposition 4

Proposition 4. If two triangles have the two sides equal to two sides respectively, and have the angles

contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal

5 to the remaining angles respectively, namely those which the equal sides subtend.

Let ABC, DEF be

two triangles having the two sides AB, AC equal to the two

10 sides DE, DF respectively, namely AB to DE and AC to DF, and the angle BAC equal to the

15 angle EDF. I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ABC to the angle DEF, and the angle ACB to the angle DFE.

For, if the triangle ABC be applied to the triangle DEF, and if the point A be placed on 20 the point D and the straight line AB on DE, then the point B will also coincide with E,

because AB is equal to DE. Again, AB coinciding with DE, the straight line AC will also coincide with DF, because

the angle BAC is equal to the angle EDF; hence the point C will also coincide with the point F, because AC is again equal to DF. But B also coincided with E; hence the base

25 BC will coincide with the base EF. [For if, when B coincides with E and C with F, the base BC does not coincide with the

base EF, two straight lines will enclose a space: which is impossible. Therefore the base BC will coincide with EF] and will be equal to it. [C.N. 4]

Thus the whole triangle ABC will coincide with the whole triangle DEF, and will be 30 equal to it. And the remaining angles will also coincide with the remaining angles and

will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle DFE.

Therefore etc. (Being) what it was required to prove.

Heath, 1956, Vol. 1, pp. 247–248, line numbers adjusted

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under what conditions the intersections will exist, or that these circles satisfy these unstated conditions. Of course, from the diagram it is clear that C exists, but the claim made in the Euclid myth is that Euclid reasons “from a small number of definitions and assumptions” not from diagrams.

We turn now to a proof that, like Liu Hui’s proofs, makes use of physical manipulations. It is Euclid’s proof of Proposition 4 of Book 1 (see Proof 3). One thing that is interesting about the proof (lines 19–32) is the lack of references to common notions, postulates or definitions. In fact, the only such reference (in line 28) is thought to be a later interpolation (Heath, 1956, p. 249). This is not surprising when one considers that the whole argument depends on the idea of picking up one triangle and putting it on top of the other one. The phrase “if the triangle ABC be applied to the triangle DEF ” (line 19) suggests that ABC be moved so that it coincides with DEF.

This way of reasoning is not what Euclid is supposed to have done, but it is quite similar to a way of reasoning used by Liu Hui:

Thus, the argumentation inevitably depends on methods. For example: ... Recourse to non-linguistic means of communication. This is necessary because, according to the adage of the Yijing cited by the commentator [Liu Hui], “not all thoughts can be adequately expressed in words” ... In place of a discourse, the reader is asked to put together jigsaw pieces, to look at a figure or to undertake calculations which themselves constitute the sole justification of the matter at hand. In each of these cases language is purely auxiliary to such procedures. (Martzloff, 1997, pp. 71–72)

In Euclid’s proof we are asked to make ABC coincide with DEF in our imaginations, and then to note the correspondences Euclid points out. This is easy to do, and quite convincing, but it is not the deductive method as described by Aristotle.

To be fair, Euclid did not reason in this way very often, and it is not, in fact, possible to deduce this proposition from his common notions, postulates and defini-tions, so he had to depart from the deductive method, or change his postulates. Hilbert took the latter approach in his Grundlagen der Geometrie (Foundations of Geometry, 1899/1921) and added this proposition as an axiom.

Reasoning from visual evidence was a mainstay of Chinese and Hindu mathe-matics, but fell out of fashion in Greece. There is evidence, however, that it was the basis for Greek mathematics as well for some time. Euclid’s proof of Proposition I.4 is part of this evidence. And, as Martzloff (1997) points out:

The Greek technical term meaning “to prove” is the verb δείκνυμι. Euclid uses this at the end of each of his proofs. Originally this verb had the precise meaning of “to point out,” “to show” or “to make visible.” Thus it appears that the Chinese proofs of Liu Hui and Li Chunfeng were similar in nature to the first known historical proofs, an example of which is given by Plato (well-known dialogue in which Socrates asks a slave how to double the area of a square); moreover, visual elements remained an essential component of proofs in China for a long time, while in Greece these were abandoned at an early stage although figurative references were maintained. (pp. 72–73)

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Martzloff’s comment on the abandonment of visual elements by the Greeks is part of the standard view of history. Netz (1998, 1999), however, has suggested that the visual element was not so much abandoned as hidden by the Greeks. He finds evidence for this in the abundance of points that are undefined except by the diagrams accompanying the proofs, and by the lettering of the diagrams which suggests that the proof was created before it was written down.

Netz describes a three stage process:

These three stages are:

(i) drawing a diagram; (ii) a dress rehearsal in front of the diagram, in which the diagram is dressed, i.e. letters are inserted (iii) a full production, writing down the proof. (1998, p. 36)

If Netz is correct, Greek proofs were basically visual and oral, with the written proof being a record of what came before. We have been equating Greek proofs with written texts, not because they were, but because all the evidence that survived was the written text.

We now turn to Euclid’s famous proof of the infinitude of primes (see Proof 4). In the Elements it has a different form from that usually given in textbooks of number theory. For an example of a modern version, see Proof 24 in Chapter 7.

Proof 4: Elements Book IX Proposition 20

PROPOSITION 20. Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime

numbers; I say that there are more 5 prime numbers than A, B, C. For let the least number measured by

A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF is either prime or not. First, let it be prime; then the prime numbers A, B, C, EF have been found which are 10 more than A, B, C. Next, let EF not be prime; therefore it is measured by some prime number. [VII. 31] Let it be measured by the prime number G. I say that G is not the same with any of the

numbers A, B, C. For, if possible, let it be so. Now A, B, C measure DE; therefore G also will measure 15 DE. But it also measures EF. Therefore G, being a number, will measure the remainder,

the unit DF: which is absurd. Therefore G is not the same with any one of the numbers A, B, C. And by hypothesis it

is prime. Therefore the prime numbers A, B, C, G have been found which are more than the 20 assigned multitude of A, B, C. Q. E. D.

Heath, 1956, Vol. 2, p. 413, line numbers adjusted

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There are two things of interest about this proof. First, the diagram is not needed, but it is there anyway. This is true in general of Euclid’s proofs in what we would now call number theory. Netz (1998) explains the presence of diagrams in these contexts. He notes that the Greek word “diagramma” refers to more than “a diagram.”

It means something much closer to “a proposition” or “a proof” (see Knorr, 1975, pp. 69–75). This is a notorious fact about Greek practice: it is generally difficult to tell whether the authors speak about drawing a figure or proving an assertion, and this is because the same words are used for both. And this again is because the diagram is the proof, it is the essence of the proof for the Greek, the metonym of the proof. (Netz, 1998, pp. 37–38)

So while Greek proofs are often taken as the model of modern discursive proofs, for the Greeks themselves they were fundamentally, essentially, associated with pictures. This explains why diagrams were included in proofs like Proof 4 which from a modern perspective do not need diagrams. To the Greeks, if it did not have a diagram, it was not a proof. The presence of diagrams where they are not needed may also be related to the origins of proofs as visual arguments, noted above.

The second thing of interest about this proof is that it does not, strictly speaking, establish that there are an infinite number of prime numbers. What it shows is that if there are three prime numbers, then there must be four prime numbers. This specific case is used to stand for all cases, a technique which is also common in Chinese proofs.

Thus, the argumentation inevitably depends on methods. For example: ... Passage from the particular to the general, based on a specific, well-chosen example. (Martzloff, 1997, p. 71)

This method of proving, known as using a generic example (see Chapter 7), is unavoidable if one does not have a method of representing unspecified numbers symbolically. Euclid had one such method, representing a number as the length of a line segment, but he did not have a method for representing an unspecified number of numbers as he had to do in this proof.

To summarise, Euclid’s proofs do not rigourously use deductive reasoning to derive propositions from axioms (common notions, postulates), definitions, and previously established propositions. They use implicit axioms, non-verbal arguments, and generic examples. This undermines the claim that they establish the proposi-tions they prove with certainty. Nonetheless, they have been the model and measure of proofs in the Western mathematical tradition for thousands of years. In Chapter 5 we will revisit the role of proving in mathematics and explore some reasons why the flaws in Euclid’s proofs were not considered serious (or even noticed) until the beginning of the twentieth century, and why they are still being offered as model mathematical proofs (e.g., by Hanna & Barbeau, 2002).

The Twentieth Century: Formalism to the Rescue?

The standard view of the history of proof acknowledges that there were some diffi-culties with less than rigourous proofs at times, especially in the seventeenth century.

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But the severity of these difficulties is sometimes minimised, and the success of the efforts to solve them overrated. According to the standard view, even if Euclid’s proofs are flawed, they were a step in the right direction, and since the work of the formalists in the early twentieth century, mathematics has once again been placed on firm foundations. Now, in principle, any mathematical proof can be expressed in purely formal statements, which can then be checked mechanically, with no chance of error due to missing assumptions, unclear definitions, use of diagrams, or logical mistakes. But this is never done, and not only for pragmatic reasons.

An ordinary page of mathematical exposition may occasionally consist entirely of mathematical symbols. To a casual eye, it may seem that there is little difference between such a page of ordinary mathematical text and a text in a formal language. But there is a crucial difference which becomes unmistakable when one reads the text. Any steps which are purely mechanical may be omitted from an ordinary mathematical text. It is sufficient to give the starting point and the final result. The steps that are included in such a text are those that are not purely mechanical — that involve some constructive idea, the intro-duction of some new element into the calculation. To read a mathematical text with understanding, one must supply the new idea which justifies the steps that are written down. (Davis & Hersh, 1981, p. 139)

The missing steps would first have to be supplied before the proof could be formalised. This would have to be done by an expert in the field, and even if an expert could be found with the patience for such a task, there would be no guarantee that the translation of the proof into a formal language would be free of error. The problem of checking the correctness of the proof becomes the problem of checking the correctness of the translation into formal language, and that is not formalisable.

The actual situation is this. On the one side, we have real mathematics, with proofs which are established by “consensus of the qualified.” A real proof is not checkable by a machine, or even by any mathematician not privy to the gestalt, the mode of thought of the particular field of mathematics in which the proof is located. Even to the “qualified reader,” there are normally differences of opinion as to whether a real proof (i.e., one that is actually spoken or written down) is complete and correct. These doubts are resolved by communication and explanation, never by transcribing the proof into first-order predicate calculus. Once a proof is “accepted,” the results of the proof are regarded as true (with very high probability). It may take generations to detect an error in a proof. If a theorem is widely known and used, its proof frequently studied, if alternate proofs are invented, if it has known applications and generalisations and is analogous to known results in related areas, then it comes to be regarded as “rock bottom.” In this way, of course, all arithmetic and Euclidean geometry are rock bottom.

On the other side, to be distinguished from real mathematics, we have “meta-mathematics” or “first-order logic.” As an activity, this is indeed part of real mathematics. But as to its content, it portrays a structure of proofs which are

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indeed infallible “in principle.” We are thereby able to study mathematically the consequences of an imagined ability to construct infallible proofs (Davis & Hersh, 1981, pp. 354–355)

This leaves us with two sorts of mathematics: first-order logic in which it is possible to begin with formal axioms and definitions and derive results using formalised logical rules without fear of error, and what Davis and Hersh call “real mathematics” in which we begin with axioms and definitions that are expressed in a mixture of formal and informal language, and derive results using informal logical rules, with an ever present possibility of error. In the first sort proofs are formal, while in the second they are semi-formal. One thing these two sorts of mathematics share with each other, and with the standard view of Euclid’s Elements, is starting from axioms and definitions that are established beforehand, and on which the truth (whether absolute or relative, certain or probable) rests. This aspect of the standard view has also had its critics, as we will see in the next section.

Lakatos and the Retransmission of Falsity

Lakatos (1961, 1976) describes the process by which he claims mathematics is discovered. In his work he criticises the Euclidean structure of definitions, postulates, theorems and proofs and the formalist reduction of mathematics to formal logic.

The history of mathematics and the logic of mathematical discovery, i.e. the phylogenesis and the ontogenesis of mathematical thought, cannot be developed without the criticism and ultimate rejection of formalism. (1976, p. 4)

We will concentrate here on Lakatos’s critique of one aspect of the standard view of proof, the claim that proofs are based on axioms and definitions that are established beforehand, that it is the (assumed) truth of the axioms that is the basis for the claimed truth of theorems. As Lakatos notes, this basing of theorems on axioms is reflected in the presentation of mathematics in textbooks and journals.

Euclidean methodology has developed a certain obligatory style of presentation. I shall refer to this as ‘deductivist style’. This style starts with a painstakingly stated list of axioms, lemmas and/or definitions. The axioms and definitions frequently look artificial and mystifyingly complicated. One is never told how these complications arose. The list of axioms and definitions is followed by the carefully worded theorems. These are loaded with heavy-going conditions; it seems impossible that anyone should ever have guessed them. The theorem is followed by the proof. (p. 142)

According to Lakatos, the order in this presentation is almost entirely the reverse of mathematical practice. Rather than beginning with axioms and definitions, he says, mathematicians begin with conjectures. After a conjecture comes a proof, but a proof is not a guarantee of the truth of the conjecture. Instead it is “a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas” (p. 127). Proving is a means of analysing the conjecture, part of a

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process he calls ‘proof-analysis’. The next stage in the process is the emergence of counterexamples to the conjecture. These counterexamples can reveal problematic definitions and hidden assumptions. Lakatos divides them into three types: The first is a counterexample to some step of the proof, but not to the conjecture itself (It is local but not global.). The second is a counterexample to some step of the proof, and to the conjecture (It is both local and global.). The third type does not contradict any step of the proof, and yet it is a counterexample to the conjecture (It is global but not local.). Each type plays a different role in the proof-analysis (p. 43). A first-type counterexample signals that there is a problem with the proof; either a hidden assumption must be revealed, or a definition changed, or a new proof produced. A second-type counterexample is the most important type for proof-analysis. When a second type counterexample emerges, the next step is to re-examine the proof to locate the step to which it is a local counterexample, the “guilty lemma”.

This guilty lemma may have previously remained “hidden” or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem — the improved conjecture — supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature. (p. 127)

The process of proof-analysis is not primarily about proving the conjecture that was its beginning, but rather improving the definitions and axioms on which it is meant to be based. “Proof-generated concepts” are important original contributions to mathematics. They account for the facts that “axioms and definitions frequently look artificial and mystifyingly complicated” and that theorems “are loaded with heavy-going conditions” (p. 142).

Counterexamples of the third type exist only if the proof analysis is invalid.

A proof-analysis is ‘rigorous’ or ‘valid’ and the corresponding mathematical theorem true if, and only if, there is no ‘third-type’ counterexample to it. I call this criterion the Principle of Retransmission of Falsity because it demands that global counterexamples be also local: falsehood should be retransmitted from the naive conjecture to the lemmas, from the consequent of the theorem to its antecedent. (p. 47)

The Principle of Retransmission of Falsity is very important to Lakatos’s thinking, and sums up what may be the most important critique in his work of the standard view of proof. In the standard view, truth is transmitted from axioms to increasingly complicated theorems. Lakatos claims that this is impossible, but more importantly, that this does not reflect the way mathematics really works. Mathematics progresses by the retransmission of falsity from conjectures to axioms and definitions. In this way counterexamples to conjectures reveal problems with the axioms and definitions. Many concepts in mathematics have existed since before the time of Euclid, but these concepts are now much more sophisticated, because conjectures based on them turned out to give rise to counterexamples which forced (because of the Principle of Retransmission of Falsity) changes to be made to the concepts.

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According to Lakatos, Euclid’s presentation of geometry seriously distorted the nature of mathematical discovery and the role of proof in it by inverting the order of things and hiding the importance of conjectures and counterexamples.

One should not forget that while proof-analysis concludes with a theorem, the Euclidean proof starts with it. In the Euclidean methodology there are no conjectures, only theorems. (p. 107, footnote 3)

This Euclidean version of proof is an integral part of the standard view of the history of proof, and so it has had significant impacts on the teaching of proof, which we will discuss in later chapters.

SUMMARY

In this chapter we have summarised what we call the standard view of the history of proof, and described some important limitations and flaws of this view. Most notably:

While many sources claim that proof originated in Greece and was not a part of the intellectual activity of other cultures, there is clear evidence of proving in ancient China and India, and it is possible that proving was part of mathematics elsewhere, in spite of the absence of evidence.

Euclid’s proofs are said to be models of rigour, however they make use of unstated assumptions and evidence from diagrams.

It is believed that mathematical proofs are (or can be made) formal, and that this means they are absolutely rigourous. In fact, formalisation of most proofs is not possible, and proofs can only be checked by a “qualified reader”.

Proofs are said to transmit truth from established axioms to the theorems they prove, but as Lakatos points out, the process can go the other way; proofs allow us to locate hidden assumptions and flawed axioms by retransmitting falsity from a conjecture with counterexamples to the underlying definitions and axioms.

Euclid himself could never have imagined the consequences of his effort to systematise the mathematics known in his day, and so it is unfair to blame him for the confusion resulting from the standard view of proof. As his name keeps coming up, however, it is convenient to use labels like Davis and Hersh’s “Euclid myth” and Lakatos’s “Euclidean methodology” to describe this point of view. And as long as we are clear that we are speaking of a particular perspective, held by many people, even today, and not of a long dead mathematician, we would agree with Lakatos that:

Euclid has been the evil genius particularly for the history of mathematics and for the teaching of mathematics, both on the introductory and the creative levels. (p. 140)

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CHAPTER 2

USAGES OF “PROOF” AND “PROVING”

The words “proof ” and “proving” are used in everyday life, mathematics, and mathe-matics education in a number of distinct ways, usually without comment. For resear-chers in mathematics education this can lead to confusion and may be a serious obstacle to future research (Balacheff, 2002/2004; Reid, 2005). Without trying to establish the “right” usages of these words, we will outline here some frequent ones and describe the differences between them.

As you read this chapter you may want to reflect on these questions: – What does “proof ” mean to you? – What should “proof ” mean to students in schools? – How can you determine what an author means by “proof ”?

EVERYDAY USAGES

In everyday English, “proof ” and “proving” can refer to convincing someone of something, or to testing something to see if it is correct.

Convincing

When we doubt a statement, we may ask, “Do you have any proof of that? Can you prove it?” In these questions proof means evidence, and proving means convincing. When Shakespeare’s Othello says, “Be sure of it; give me the ocular proof ” (Act III, scene 1) he means that Iago must convince him of the truth of his accusation by providing visible evidence. What counts as convincing evidence depends on context, and may include physical force, verbal abuse, social pressure, or anything else that persuades someone else. In the Sidney Harris cartoon captioned “You want proof ? I’ll give you proof !” the humour comes from a shift in context, as one mathematician is shown convincing another mathematician by punching him in the nose, which is an everyday, but not a mathematical usage of “proof ” as convincing.

Testing

“Prove” is derived from the Latin verb probare, which means to test, to try. The English verb “probe” still carries this meaning. Taking “prove” as meaning “con-vince” when it means “test” can lead to odd interpretations of common expressions. For example, the expression “the exception which proves the rule” is often taken in the paradoxical sense of asserting that the presence of a counterexample establishes

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the general truth of a rule, which follows if “prove” is taken to mean providing convincing evidence. However, the expression is not so paradoxical if “prove” is being used to mean “test”. Then saying “the exception proves the rules” amounts to suggesting that examining exceptions closely and reasoning out the way they occur can lead to a clarification and improvement of the rule. This interpretation is reminiscent of Lakatos’s (1976) process of proof-analysis in which counter-examples and proving interact to improve theorems in mathematics (see Chapters 1 and 11).

The use of “prove” to mean “test, try” can also occur in the noun form; a “proof ” can be a test or a trial. In some common phrases, “proof-read,” “proof of the pudding,” “100 proof,” the word “proof ” is used in this way. Words like “waterproof” and “fireproof” are also based on this meaning; they describe objects that have been tested and found to be resistant.

SCIENTIFIC USAGES

When one reads an article about a scientific discovery, one might encounter the words “proof ” and “proving” used to refer to convincing, but on the basis of special types of evidence.

Experiments Prove Existence Of Atomic Chain ‘Anchors’

Atoms at the ends of self-assembled atomic chains act like anchors with lower energy levels than the “links” in the chain, according to new measure-ments by physicists at the National Institute of Standards and Technology (NIST).

The first-ever proof of the formation of “end states” in atomic chains may help scientists design nanostructures, such as electrical wires made “from the atoms up,” with desired electrical properties. (NIST, 2005, italics added)

When scientists “prove” something they offer convincing evidence, but that evidence must be of a special type appropriate to science.

MATHEMATICAL USAGES

Godino and Recio (1997, Recio & Godino, 2001) make a distinction between two usages of the words “proof ” and “proving” in two areas of mathematics: foun-dations of mathematics and mainstream mathematics. This distinction is similar to the distinction made by Douek (1998) between “formal proofs” and “mathematical proofs”, the distinction made by Davis and Hersh (1981) between metamathematics and “real mathematics”, and our distinction between formal proofs and semi-formal proofs which we mentioned in Chapter 1.

In foundations of mathematics, proofs give theorems “a universal and intemporal validity”, “they rest on the validity of the logic rules used,” “the use of formal lang-uages is required,” and proving is a way of coming to grips “with the theoretical

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problem of organizing and structuring the system of mathematical knowledge” (Godino & Recio, 1997, pp. 315–316). Hanna (1983) defines formal proofs in this way:

The term rigorous proof or formal proof ... is understood here to mean a proof in mathematics or logic which satisfies two conditions of explicitness. First, every definition, assumption, and rule of inference appealed to in the proof has been, or could be, explicitly stated; in other words, the proof is carried out within the frame of reference of a specific known axiomatic system. Second, every step in the chain of deductions which constitutes the proof is set out explicitly. (p. 3)

In mainstream professional mathematics, theorems do not have the “character of absolute and necessary truths”, the validity of proofs is “‘judged by qualified judges’ (Hersh, 1993, p. 389)”, “proofs are deductive but not formal,” and proving is a way “to solve new problems, to increase the knowledge body, and, secondarily, to organize and found the whole system of mathematics” (Godino & Recio, 1997, pp. 316–317).

Mathematicians working in the foundational domain of proof theory recognise this distinction, and use “formal proof” to refer to the proofs they study, and “social proof ” to refer to the proofs of mainstream mathematicians. In Chapter 1 we suggested the adjective “semi-formal” to refer to these mainstream proofs. Davis and Hersh (1981) note that although an “ideal” mainstream mathematician might claim his semi-formal proofs meet the same criteria as those of mathematicians working on foundations, when pressed, he would admit the differences:

What you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof. ... Oh, of course no one ever really does it. It would take forever! ... [A proof is really] an argument that convinces someone who knows the subject. (pp. 39–40)

USAGES IN MATHEMATICS EDUCATION RESEARCH

Many researchers in mathematics education use the words “proof ” and “proving”, in a number of distinct ways. Most use the words in different ways within the same paper.

For example, consider this sentence:

Even when students seem to understand the function of proof in the mathe-matics classroom ... and to recognise that proofs must be general, they still frequently fail to employ an accepted method of proving to convince themselves of the truth of a new conjecture, preferring instead to rely on pragmatic methods and more data. (Hoyles & Küchemann, 2002, p. 194, references removed for clarity, italics added)

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The use of the singular form “proof ” in the first line instead of the plural “proofs” suggests that the word is being used to mean a concept or category. In the second line the plural is used, suggesting that a set of objects is meant. Finally in the third line, the verb “proving” is used. The fact that “method of proving” seems to include “accepted methods” as well as “pragmatic methods” suggests that “proving” is used to mean something different than “constructing a proof” in this case.

This suggests a starting point for an investigation of the usage of the words “proof ” and “proving” in research in mathematics education. Three categories of usage can be distinguished on purely grammatical grounds:

1) The use of “proof ” in the singular, without an article to refer to a concept. 2) The use of “proof ” with an article or in the plural to refer to an object. 3) The use of the verb “prove” to refer to an action or process.

Note that “proving” is a difficult case, as it can be a form of the verb “prove” but also a noun: “Jim is proving the theorem” or “Jim’s proving of the theorem”.

Considering word usage in mathematics education research even at the surface level of the forms of words reveals some striking differences. For example, consider the frequency of the use of the words “proof ”, “proofs”, “prove” (including “proves” “proven” and “proved”), “proving”, words beginning with “argu+” (“argument”, “arguing”, “argue”, etc.) and “reasoning”. In Figure 2 the frequency of the usage of these words in three papers published in Educational Studies in Mathematics is shown. The left hand column shows an example of an author (Fischbein, 1999) who uses the verb “prove” more often than the nouns “proof ” and “proofs”. In contrast, the right hand column shows an example of an author (Uhlig, 2002) who uses the nouns much more than the verb. It is clear from the centre column that Hanna (2000) uses the word “proof ” much more than “prove”, but it is not clear whether she means a concept or an object when she writes “proof ”. A closer look at the article clarifies this. Hanna’s use of “proof ” breaks down into four categories: – “proofs” in the plural form, 32 occurrences, 18% – “proof ” preceded by “a”, 17 occurrences, 10% – “proof ” preceded by “the”, 7 occurrences, 4% – other uses of “proof ”, 122 occurrences, 71%

The final category still contains a few uses of “proof ” to refer to an object (for example when it is preceded by an adjective, e.g., “an explanatory proof”), but most uses of the word refer to a concept.

In the following we will go into more detail about the ways mathematics education researchers use “proof ” and “proving” to refer to a concept, an object, or a process. Note, however, that we do not claim that any researcher’s usages fall neatly into a single category, nor that the usages we describe here are themselves disjoint categories. As the quote from Hoyles and Küchemann at the start of this section indicates, several usages can occur in a single paragraph. And while the three ESM articles analysed in Figure 2 show the predominance of some usages over others, almost all usages appear in all three articles.

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Figure 2. Word usage in three ESM papers

Proof as a Concept

The use of the word “proof ” to refer to a concept is usually clear from the context or from syntactical considerations, but once one knows that the word is intended to refer to a concept, does one know to what concept it is meant to refer? Unfortunately, no. Researchers in mathematics education have a wide range of perspectives on proof which make it difficult to know what concept they might mean by the word “proof ”. In the next chapter we will describe some researchers’ perspectives, but as many researchers do not provide enough clues in their writing to definitively identify their perspectives, we can only leave the reader with the advice to be wary.

Proof as an Object

There are a number of different objects “proofs” can refer to in mathematics education research, and those objects can be distinguished by their forms or by their function. The two most common usages are to refer to texts, usually written texts, of a certain form, or to refer to arguments, spoken or written, with the function of convincing.

Proof-Texts. The majority of the [high attaining 14 and 15 year old] students were unable to construct valid proofs in [the domain of number and algebra]. (Healy & Hoyles, 2000, p. 425)

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In mathematics schoolbooks and journals, one encounters some texts under the heading “proof ”. Such proof-texts are characterised by a particular form and style. The proof-texts of schoolbooks are different from the proof-texts of professional mathematics journals, but there is sufficient unity in the styles to justify the use of the same term for both. Writing proof-texts is a goal of recent reform documents: “High school students should be able to present mathematical arguments in written forms that would be acceptable to professional mathematicians” (National Council of Teachers of Mathematics [NCTM], 2000, p. 58). That students cannot do this is what is meant when researchers such as Duval (1990), Senk (1985) and Healy and Hoyles (2000) conclude that students do not understand proof.

“Proving” can refer to writing a proof-text (e.g., Douek, 1998) but not everyone who uses “proof ” to refer to proof-texts uses “proving” in this way.

Convincing arguments. The use of “proof ” to refer to a convincing argument by mathematics education researchers is essentially a return to the everyday usage we noted above. But the audience to be convinced can vary. For example, for Mason, Burton, and Stacey (1982) a proof is an argument that convinces an enemy, for Davis and Hersh (1981) it is an argument that convinces a mathematician who knows the subject and for Volmink (1990) it is an argument that convinces a reasonable sceptic. In all cases, it is not the argument itself that makes it a proof, but rather that fact that it convinces someone. As Manin points out, in this often cited quotation, “A proof becomes a proof after the social act of ‘accepting it as a proof’. This is true of mathematics as it is in physics, linguistics, and biology.” (1977, p. 48). Using proof this way necessarily means whether a given proof-text is a proof or not can vary. “Proof is that which compels belief. That means that proof is different in different eras, and indeed, that it is different for different people at any one time.” (MacKernan, 1996, p. 14).

If “proof ” is used to mean a convincing argument, then “proving” usually refers to convincing someone of something. This usage is compatible with some ways of using “proof ” is used to refer to a reasoning process or a social discourse (see below).

Proof as a Process

“Proof ” can refer to a psychological process of reasoning, or to a social process, a certain kind of discourse. In both cases, (as with proof as an object) what process is being referred to can be determined both by the form of the process, and by its function.

Deductive reasoning. “In fact, ‘proof’ is just ‘reasoning’, but careful, critical reasoning looking closely for gaps and exceptions” (Hersh, 2009, p. 19). When “proof ” refers to deductive reasoning it is being used to refer to a psychological process which takes on a certain form. Because psychological processes are not directly observable, specifying this form is difficult. It can be loosely described as a chain or tree of connected statements beginning from some that are taken as true

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and proceeding to a conclusion according to a few logical rules (modus ponens, etc.) occasionally supplemented by special rules unique to mathematics (e.g., the principle of mathematical induction; see Chapter 6 for details). Some authors (e.g., Reid 1995b) use the verb form “proving” to refer to reasoning deductively, but use the noun form “proof ” in another way.

Reasoning for a purpose. For some researchers, “proof ” refers to a reasoning process, but not necessarily a deductive one. For Harel and Sowder, for example, “the emphasis is on the student’s thinking rather than on what he or she writes” (Harel & Sowder, 1998, p. 276) but it is not the nature of the reasoning process that is important, but instead the function that it serves.

By “proving” we mean the process employed by an individual to remove or create doubts about the truth of an observation. (p. 241)

In Harel and Sowder’s case, that function is verification of the truth of a statement. They are careful to distinguish between “proving”, “proof ” and what they call “proof schemes” but all are related to the purpose of verifying. “Proving” is a mental act, “the process of removing or instilling doubts about an assertion” (Harel, 2007, p. 65). “Proof ” is “a particular statement one offers to ascertain for oneself or convince others” (p. 66), a proof-text. A “proof scheme” is a characteristic “way of thinking associated with the proving act” (p. 66).

Another function of proof as a process of reasoning is understanding. Raman seems to use “proof ” in this way, when referring to “the private aspect of proof”:

I distinguish between a private and a public aspect of proof, the private being that which engenders understanding and provides a sense of why a claim is true. The public aspect is the formal argument with sufficient rigor for a particular mathematical setting which gives a sense that the claim is true. (Raman, 2002, p. 3; see also 2003, p. 320)

Raman’s “public aspect of proof” seems to refer to proof as an object, either proof-texts or convincing arguments.

Discourse defined by function. Knipping (2004, p. 73) discusses “collective proving processes” or “argumentations” which are “collective processes in which students and teacher develop the proof together.” The “collective proving process” she refers to are embedded in a “proving discourse”. This discourse is a social process whose function is producing reasons for the truth of a statement.

Balacheff (1988b) seems to use “proof ” in the same sense when he writes:

Le passage de l’explication à la preuve fait référence à un processus social par lequel un discours assurant la validité d’une proposition change de statut en étant acceptée par une communauté [The transition from explanation to proof refers to a social process by which a discourse asserting the truth of a proposition, gives it the status of being accepted by the community.] (p. 29)

Consider, for example, a class attempting to decide the truth of the proposition “The centre of gravity of a triangle is at the intersection of the medians”. Conjectures

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are voiced and reasons given. Through a process of social negotiation (probably guided in significant ways by the teacher) an argument is produced that verifies the truth of the proposition. Note that the criteria for accepting an argument depend on the class (including the teacher) or more generally on the community. Arguments might be accepted by some community that would be rejected by others.

Discourse defined by form: Deductive discourse. When deductive reasoning is expressed in a social context it becomes a method of arguing. Above we noted that the kind of arguments accepted by a community is community dependent. Some communities, notably communities of mathematicians, insist on a deductive basis for acceptable arguments. In such communities if “proving” refers to a collective process, it will only be used to describe processes with a deductive basis.

To return to the example of the centre of gravity of a triangle, a possible argument would make use of cut-out triangles of various sizes and angles and empirical testing of the locations of their centres of gravity. Such an approach might be acceptable in a physics classroom, but not in a mathematics classroom. In the mathematics classroom the argument would have to proceed deductively, perhaps by establishing that each median divides the triangle into two equal areas, that the three medians meet in a single point, and finally that any line through this point will divide the triangle into equal areas (However, see Proof 10 in Chapter 6 for a proof that bridges these two contexts).

“DEMONSTRATION” AND “PROOF ” IN OTHER LANGUAGES

As the research literature on proof in mathematics education includes significant contributions in languages other than English it is worth being aware of some issues related to word usage in other languages.

In older English language texts on proof and proving in mathematics education (e.g., Fawcett, 1938) one encounters the word “demonstration” used to refer to mathe-matical proofs. This word is now rarely used in this sense, more often being used to refer to a political protest or a presentation intended to show how something works. In Romance languages, however, cognate words (e.g., démonstration, dimostrazione, demostración) continue to be used, and Balacheff, for example, makes a distinction between “démonstration” and “preuve” (the French cognate of “proof ”).

Nous appelons preuve une explication acceptée par une communauté donnée à un moment donné. ... Au sien de la communauté mathématique ne peuvent être acceptées pour preuve que des explications adoptant une forme particulière. Elles sont une suite d’énoncés organisée suivant des règles déterminées: un énoncé est connu comme étant vrai ou bien est déduit de ceux qui le précèdent à l’aide d’une règle de déduction prise dans un ensemble de règles bien défini. Nous appelons démonstrations ces preuves. (1987, p. 148)

[We call proof an explanation accepted by a given community at a given moment... Within the mathematical community only explanations adopting a particular form can be accepted as proofs. They are an organized succession

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of statements following specified rules: a statement is known to be true or is deduced from those which precede it using a deductive rule taken from a well defined set of rules. We call such proofs “démonstrations”. ]

When writing in English Balacheff attempts to preserve this distinction by translating “démonstration” as “mathematical proof”.

By “proof ” we mean a discourse whose aim is to establish the truth of a conjecture (in French: Preuve), not necessarily a mathematical proof (in French: Démonstration) (Balacheff, 1991b, p. 109, Note 2)

But most English writers do not use “proof ” and “mathematical proof” in the same way as Balacheff does, and many authors writing in French, Italian and Spanish do not make the same distinction between “preuve”, “prova” and “prueba” and “démonstration”, “dimostrazione”, and “demostración”.

In German the situation is like that of English. “Proof ” can be translated as “Beweis” and the German word “Demonstration” means approximately the same thing as the English word “demonstration”.

SUMMARY

The words “proof” and “proving” can be used in a number of ways, even in an academic discipline like mathematics education where the exact meanings of these words would seem to be important. As Herbst and Balacheff (2009) note, ignoring these multiple usages can lead to a deadlock in efforts to communicate. But the answer is not to insist on one “correct” usage.

If the field is in a deadlock as regards to what we mean by “proof,” we contend this is so partly because of the insistence on a comprehensive notion of proof that can serve as referent for every use of the word. ... We have argued that to make it operational for understanding and appraising the mathe-matics of classrooms we need at least three meanings for the word. (p. 62)

We have identified a number of usages in this chapter: – A concept of proof – Proof-texts – Convincing arguments – Deductive reasoning – Personal verification – Personal understanding – A social discourse to verify – A deductive social discourse

These usages are not disjoint categories, nor does a researcher’s use of “proof ” or “proving” in one way in one context guarantee that her or his next usage will be the same. However, being aware that there are different usages is an important step to being able to decipher mathematics education research.

In our writing we will attempt to use more precise words to say what we mean, reserving the word “proof ” primarily to refer to a concept. However, to avoid unnecessary repetitions we may use “proof ” and “proving” in one of their

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other senses when the meaning is clear. Similarly, when quoting others we will clarify how these words are being used if possible and necessary. If the meaning is sufficiently clear from the context, or if the meaning is so unclear we cannot determine what it is, we will not attempt to suggest how the author is using “proof ” and “proving”.

Word usages can also offer important hints towards larger issues. In the next chapter we will use three of these usages, proof-texts, reasoning, and discourse, to distinguish between theoretical perspectives in mathematics education research.